May 21, 2024

Ulam’s spirals reveal the regularity of prime numbers

Diagonal in the vortex Olam | All prime numbers are in the form 4x2 – 2x + 41 is highlighted in blue.

In general, the diagonals in an Ulam spiral can be described by the first formula. Obviously, the function does not work with any arbitrary set of a B C And n (which all contain integer values) Returns a prime number. If that were the case, one would have solved one of the biggest unanswered questions in mathematics: that of the distribution of prime numbers. However, the Spiral of Ulam shows that there are certain groups of far And c There, for the one with the job and (n) It can calculate many more prime numbers than other combinations.

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Ulam found a few of these combinations himself; Other mathematicians also frequently dealt with Ulam spirals (as they are now called). However, there is still no definitive explanation for the pattern in the spirals.

Clubber’s Triangle | Prime numbers in the form of x2 – x + 41 in orange.

Stanisław Ulam was not the first to come across a phenomenon of this kind. As early as 1932, American herpetologist Lawrence Klauber arranged the natural numbers as a triangle, with 1 at the top. If you mark prime numbers there, they are most often on diagonal and vertical lines. Herpetology, by the way, is not a mathematical subject, but the science of amphibians and reptiles – Klopper was one of the leading experts on rattlesnakes. Not necessarily the kind of research most likely to yield discoveries about prime numbers, but creativity is a bit like love in this respect: it sees, to paraphrase Shakespeare, with the mind rather than the eyes—and meets mathematicians and rattlesnake researchers alike.

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